{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:01:06Z","timestamp":1760151666279,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T00:00:00Z","timestamp":1649030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a \u201cHamiltonian\u201d and an entropy, playing the role of dynamics generator. Generally, these tensors are a Poisson bracket tensor, describing the Hamiltonian part of the dynamics, and a symmetric metric tensor, that models purely dissipative dynamics. In this paper, the metriplectic system describing a simplified two-photon absorption by a two-level atom is disclosed. The Hamiltonian component is sufficient to describe the free electromagnetic radiation. The metric component encodes the radiation\u2013matter coupling, driving the system to an asymptotically stable state in which the excited level of the atom is populated due to absorption, and the radiation has disappeared. First, a description of the system is used, based on the real\u2013imaginary decomposition of the electromagnetic field phasor; then, the whole metriplectic system is re-written in terms of the phase\u2013amplitude pair, named Madelung variables. This work is intended as a first result to pave the way for applying the metriplectic formalism to many other irreversible processes in nonlinear optics.<\/jats:p>","DOI":"10.3390\/e24040506","type":"journal-article","created":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T09:49:41Z","timestamp":1649065781000},"page":"506","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Metriplectic Structure of a Radiation\u2013Matter-Interaction Toy Model"],"prefix":"10.3390","volume":"24","author":[{"given":"Massimo","family":"Materassi","sequence":"first","affiliation":[{"name":"Institute for Complex Systems, National Research Council (ISC-CNR), Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy"},{"name":"Institute for Space Astrophysics and Planetology (IAPS), National Institute of Astrophysics (INAF), 00133 Rome, Italy"}]},{"given":"Giulia","family":"Marcucci","sequence":"additional","affiliation":[{"name":"Apoha Ltd., 242 Acklam Rd., London W10 5JJ, UK"},{"name":"Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy"},{"name":"Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy"}]},{"given":"Claudio","family":"Conti","sequence":"additional","affiliation":[{"name":"Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy"},{"name":"Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"052136","DOI":"10.1103\/PhysRevA.94.052136","article-title":"Irreversible evolution of a wave packet in the rigged-Hilbert-space quantum mechanics","volume":"94","author":"Marcucci","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1002\/prop.201200077","article-title":"Amplification of non-Markovian decay due to bound state absorption into continuum","volume":"61","author":"Garmon","year":"2013","journal-title":"Fortschr. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/0167-2789(86)90209-5","article-title":"A paradigm for joined Hamiltonian and dissipative systems","volume":"18","author":"Morrison","year":"1986","journal-title":"Physica D"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/0375-9601(87)90185-X","article-title":"Fetal alcohol spectrum disorder: Potential role of endocannabinoids signaling","volume":"125","author":"Turski","year":"1987","journal-title":"Phys. Lett. A"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Petru, Z., Przystawa, J., and Rapcewicz, K. (1996). From Quantum Mechanics to Technology, Springer.","DOI":"10.1007\/BFb0106010"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Oettinger, H.C. (2004). Beyond Equilibrium Thermodynamics, Wiley.","DOI":"10.1002\/0471727903"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"45","DOI":"10.32370\/IA_2012_07_3","article-title":"Algebrizing friction: A brief look at the Metriplectic Formalism","volume":"1","author":"Materassi","year":"2012","journal-title":"Intellect. Arch."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/0375-9601(84)90635-2","article-title":"Bracket formulation for irreversible classical fields","volume":"100","author":"Morrison","year":"1984","journal-title":"Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.physd.2011.12.013","article-title":"Metriplectic framework for dissipative magneto-hydrodynamics","volume":"241","author":"Materassi","year":"2012","journal-title":"Physica D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"78","DOI":"10.35470\/2226-4116-2018-7-2-78-86","article-title":"Metriplectic torque for rotation control of a rigid body","volume":"7","author":"Materassi","year":"2018","journal-title":"Cybern. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.jmaa.2006.02.023","article-title":"Metriplectic structure, Leibniz dynamics and dissipative systems","volume":"326","author":"Guha","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Materassi, M. (2016). Entropy as a metric generator of dissipation in complete metriplectic systems. Entropy, 18.","DOI":"10.3390\/e18080304"},{"key":"ref_13","unstructured":"Boyd, R.W. (2008). Nonlinear Optics, Academic Press. [3rd ed.]."},{"key":"ref_14","unstructured":"Moloney, J., and Newell, A. (2004). Nonlinear Optics, Advanced Book Program, Avalon Publishing."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1109\/3.552252","article-title":"Theory and simulation on the threshold of water breakdown induced by focused ultrashort laser pulses","volume":"33","author":"Feng","year":"1997","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1364\/OL.23.000382","article-title":"Dynamic spatial replenishment of femtosecond pulses propagating in air","volume":"23","author":"Mlejnek","year":"1998","journal-title":"Opt. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/S0030-4018(00)00731-8","article-title":"Ultraviolet filamentation in air","volume":"180","author":"Schwarz","year":"2000","journal-title":"Opt. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Materassi, M. (2015). Metriplectic algebra for dissipative fluids in Lagrangian formulation. Entropy, 17.","DOI":"10.3390\/e17031329"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/506\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:49:40Z","timestamp":1760136580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/506"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,4]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040506"],"URL":"https:\/\/doi.org\/10.3390\/e24040506","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,4,4]]}}}